%%
%% properties.tex
%% 
%% Made by Alex Nelson
%% Login   <alex@tomato>
%% 
%% Started on  Sat May 30 18:33:16 2009 Alex Nelson
%% Last update Sat May 30 18:33:16 2009 Alex Nelson
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There are some important properties of the real numbers that we
need to introduce, since we are introducing real analysis.

\begin{prop}[Triangle Inequality]
For any $a,b\in\mathbb{R}$, we have
\begin{equation}
|a+b|\leq|a|+|b|.
\end{equation}
\end{prop}
\begin{rmk}
More generally, in metric topology, when we have a metric
$$d:X\times X\to\mathbb{R},$$ the identity
\begin{equation}
d(x,z)+d(z,y)\geq d(x,y)
\end{equation}
holds for all $x,y,z\in X$. It is called the ``triangle inequality''.
\end{rmk}
